Energy Identities for the Wave Equation
Abstract
Identities for the wave equation similar to the energy identity were previously derived by showing that if u is sufficiently smooth, there exist linear first order operators Nu such that Nu square u, with square = partial derivative exp 2/partial derivative of t exp 2 - delta can be written as the divergence of a vector. In Part I of this report these identities are rederived for three space variables by noting that certain transformations leave the wave operator invariant and hence the classical energy identity can be transformed into other identities. In part II, the Kelvin transformation and the resulting identity are applied to incoming and outgoing waves. In Part III the main theorem of the first part is used to prove the following result in geometrical optics: Suppose that we are given a smooth, star-shaped perfectly reflecting, three-dimensional body that extends to infinity and that a high frequency harmonic source of light illuminates the region outside the body in such a way that no shadow is cast. The field is given by a solution of a boundary value problem for the reduced wave equation. There is also an approximate solution given by geometrical optics. The theorem states that these two are asymptotically equal in the limit of infinite frequency for the harmonic source.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1966
- Accession Number
- AD0630506
Entities
People
- Cathleen S. Morawetz
Organizations
- New York University